Window Calculation Module¶

This section describes two potential modeling approaches for Windows.  The first (layer by layer) is implemented.  The second, simple approach, reuses the layer-by-layer approach but converts an arbitrary window performance into an equivalent single layer.

The primary Window calculation is a layer-by-layer approach where windows are considered to be composed of the following components, only the first of which, glazing, is required to be present:

• Glazing, which consists of one or more plane/parallel glass layers. If there are two or more glass layers, the layers are separated by gaps filled with air or another gas. The glazing optical and thermal calculations are based on algorithms from the WINDOW 4 and WINDOW 5 programs [Arasteh et al., 1989], [Finlayson et al., 1993].  Glazing layers are described using the input object WindowMaterial:Glazing.

• Gap, layers filled with air or another gas that separate glazing layers.  Gaps are described using the input object WindowMaterial:Gas.

• Frame, which surrounds the glazing on four sides.  Frames are described using the input object WindowProperty:FrameAndDivider.

• Divider, which consists of horizontal and/or vertical elements that divide the glazing into individual lites.

• Shading device, which is a separate layer, such as drapery, roller shade or blind, on the inside or outside of the glazing, whose purpose is to reduce solar gain, reduce heat loss (movable insulation) or control daylight glare.  Shading layers are described using "WindowShadingControl" input objects.

In the following, the description of the layer-by-layer glazing algorithms is based on material from Finlayson et al., 1993. The frame and divider thermal model, and the shading device optical and thermal models, are new to EnergyPlus.

A second approach has been developed where windows are modeled in a simplified approach that requires minimal user input that is processed to develop and equivalent layer that then reuses much of the layer-by-model.  This "Simple Window Construction: model is described below.

Optical Properties of Glazing¶

The solar radiation transmitted by a system of glass layers and the solar radiation absorbed in each layer depends on the solar transmittance, reflectance and absorptance properties of the individual layers. The absorbed solar radiation enters the glazing heat balance calculation that determines the inside surface temperature and, therefore, the heat gain to the zone from the glazing (see "Window Heat Balance Calculation"). The transmitted solar radiation is absorbed by interior zone surfaces and, therefore, contributes to the zone heat balance. In addition, the visible transmittance of the glazing is an important factor in the calculation of interior daylight illuminance from the glazing.

Table: Variables in Window Calculations

Glass Layer Properties¶

In EnergyPlus, the optical properties of individual glass layers are given by the following quantities at normal incidence as a function of wavelength:

Transmittance, T

Front reflectance, R\(^{f}\)

Back reflectance, R\(^{b}\)

Here "front" refers to radiation incident on the side of the glass closest to the outside environment, and "back" refers to radiant incident on the side of the glass closest to the inside environment. For glazing in exterior walls, "front" is therefore the side closest to the outside air and "back" is the side closest to the zone air. For glazing in interior (i.e., interzone) walls, "back" is the side closest to the zone in which the wall is defined in and "front" is the side closest to the adjacent zone.

Glass Optical Properties Conversion¶

Conversion from Glass Optical Properties Specified as Index of Refraction and Transmittance at Normal Incidence¶

The optical properties of uncoated glass are sometimes specified by index of refraction, n,* * and transmittance at normal incidence, T.

The following equations show how to convert from this set of values to the transmittance and reflectance values required by Material:WindowGlass. These equations apply only to uncoated glass, and can be used to convert either spectral-average solar properties or spectral-average visible properties (in general, n and T are different for the solar and visible). Note that since the glass is uncoated, the front and back reflectances are the same and equal to the R that is solved for in the following equations.

Given n and T, find R:

Example:

Simple Window Model¶

EnergyPlus includes an alternate model that allows users to enter in simplified window performance indices.  This model is accessed through the WindowMaterial:SimpleGlazingSystem input object and converts the simple indices into an equivalent single layer window.  (In addition a special model is used to determine the angular properties of the system -- described below).  Once the model generates the properties for the layer, the program reuses the bulk of the layer-by-layer model for subsequent calculations.  The properties of the equivalent layer are determined using the step by step method outlined by Arasteh, Kohler, and Griffith (2009) with modifications to formulate the angular performance in a manner consistent with the angular properties for coated glass in other window models.  The core equations are documented here.  The reference contains additional information.

The simplified window model accepts U and SHGC indices and is useful for several reasons:

1)   Sometimes, the only thing that is known about the window are its U and SHGC;

2)   Codes, standards, and voluntary programs are developed in these terms;

3)   A single-layer calculation is faster than multi-layer calculations.

Note: This use of U and SHGC to describe the thermal properties of windows is only appropriate for specular glazings.

While it is important to include the ability to model windows with only U-value and SHGC, we note that any method to use U and SHGC alone in building simulation software will inherently be approximate.  This is due primarily to the following factors:

• SHGC combines directly transmitted solar radiation and radiation absorbed by the glass which flows inward.  These have different implications for space heating/cooling.  Different windows with the same SHGC often have different ratios of transmitted to absorbed solar radiation.

• SHGC is determined at normal incidence; angular properties of glazings vary with number of layers, tints, coatings.  So products which have the same SHGC, can have different angular properties.

• Window U-factors vary with temperatures.

Thus, for modeling specific windows, we recommend using more detailed data than just the U and SHGC, if at all possible.

The simplified window model determines the properties of an equivalent layer in the following steps.

Step 1.  Determine glass-to-glass Resistance.¶

Window U-values include interior and exterior surface heat transfer coefficients.  The resistance of the bare window product, or glass-to-glass resistance is augmented by these film coefficients so that,

Where,

\({R_{i,w}}\) is the resistance of the interior film coefficient under standard winter conditions in units of m\(^{2}\)·K/W,

\({R_{o,w}}\) is the resistance of the exterior film coefficient under standard winter conditions in units of m\(^{2}\)·K/W, and

\({R_{l,w}}\) is the resistance of the bare window under winter conditions (without the film coefficients) in units of m\(^{2}\)·K/W.

The values for \({R_{i,w}}\) and \({R_{o,w}}\) depend on U and are calculated using the following correlations.

So that the glass-to-glass resistance is calculated using:

Because the window model in EnergyPlus is for flat geometries, the models are not necessarily applicable to low-performance projecting products, such as skylights with uninsulated curbs.  The model cannot support glazing systems with a U higher than 7.0 because the thermal resistance of the film coefficients alone can provide this level of performance and none of the various resistances can be negative.

Step 2.  Determine Layer Thickness.¶

The thickness of the equivalent layer in units of meters is calculated using,

Step 3.  Determine Layer Thermal Conductivity¶

The effective thermal conductivity, \({\lambda_{eff}}\), of the equivalent layer is calculated using,

Step 4.  Determine Layer Solar Transmittance¶

The layer's solar transmittance at normal incidence, \({T_{sol}}\), is calculated using correlations that are a function of SHGC and U-Factor.

And for U-values between 3.4 and 4.5, the value for \({T_{sol}}\) is interpolated using results of the equations for both ranges.

Step 5.  Determine Layer Solar Reflectance¶

The layer's solar reflectance is calculated by first determining the inward flowing fraction which requires values for the resistance of the inside and outside film coefficients under summer conditions, \({R_{i,s}}\) and \({R_{o,s}}\), respectively.  The correlations are:

And for U-values between 3.4 and 4.5, the values are interpolated using results from both sets of equations.

The inward flowing fraction, \(Fra{c_{inward}}\), is then calculated using:

Then, the solar reflectances of the front face, \({R_{s,f}}\), and back face, \({R_{s,b}}\), are calculated using:

The thermal absorptance, or emittance, is taken as 0.84 for both the front and back and the longwave transmittance is 0.0.

Step 6.  Determine Layer Visible Properties¶

The user has the option of entering a value for visible transmittance as one of the simple performance indices.  If the user does not enter a value, then the visible properties are the same as the solar properties.  If the user does enter a value then layer's visible transmittance at normal incidence, \({T_{Vis}}\), is set to that value.  The visible light reflectance for the back surface is calculated using:

The visible light reflectance for the front surface is calculated using:

Step 7. Determine Angular Performance¶

The angular properties of windows are important because during energy modeling, the solar incidence angles are usually fairly high.  Angles of incidence are defined as angles from the normal direction extending out from the window.  The simple glazing system model includes a range of correlations that are selected based on the values for U and SHGC.  These were chosen to match the types of windows likely to have such performance levels.  The matrix of possible combinations of U and SHGC values have been mapped to a set of 28 bins shown in the Figure 1.

There are ten different correlations, A thru J, for both transmission and reflectance.  The correlations are used in various weighting and interpolation schemes according the figure above.  The correlations are normalized against the performance at normal incidence.  EnergyPlus uses these correlations to store the glazing system's angular performance at 10 degree increments and interpolates between them during simulations.  The model equations use the cosine of the incidence angle, \(\cos (\phi )\), as the independent variable.  The correlations for transmittance have the form:

where

While the original method described by Arasteh, Kohler, and Griffith (2009) uses a similar equation form for reflectance, the form of the equation and its coefficients used in EnergyPlus have been modified algebraically to use a form consistent with what is used for coated glass elsewhere in the program. To convert from the original equation form,

to the form used for coated glazing,

set them equal to each other and solve algebraically for \(\rho_{new}(\phi )\):

where,

and,

The updated coefficient values for a, b, c, d, and e are listed for transmittance and reflectance in Tables 2 and 3, respectively.

Table: Normalized Transmittance Correlations for Angular Performance

Table: Normalized Reflectance Correlations for Angular Performance

Application Issues¶

EnergyPlus's normal process of running the detailed layer-by-layer model, with the equivalent layer produced by this model, creates reports (sent to the EIO file) of the overall performance indices and the properties of the equivalent layer. Both of these raise issues that may be confusing.

The simplified window model does not reuse all aspects of the detailed layer-by-layer model, in that the angular solar transmission properties use a different model when the simple window model is in effect. If the user takes the material properties of an equivalent glazing layer from the simple window model and then re-enters them into just the detailed model, then the performance will not be the same because of the angular transmission model will have changed. It is not proper use of the model to re-enter the equivalent layer's properties and expect the exact level of performance.

There may not be an exact agreement between the performance indices echoed out and those input in the model. This is expected with the model and the result of a number of factors. For example, when there is a conflict between the SHGC and the U that are not physical and compromises need to be made. In the versions up till 9.6.0, the reported U value is limited to no higher than about 5.8W/m\(^{2}\)\(\cdot\)K when input is allowed to go up to U-7 W/m\(^{2}\)\(\cdot\)K. In later versions, this mismatch of the input and the reported U-factors among exterior windows are resolved with the application of an adjustment ratio. The adjustment ratio is computed iteratively. In each iteration, the nominal (or effective) U is re-evaluated, and the adjustment ratio at the current iteration is computed as a ratio between the input U and nominal U at the current iteration. The iterative process stops when the input U and the nominal U is close enough (with a less than 0.01 W/m\(^{2}\)\(\cdot\)K difference). In general, the simple window model is intended to generate a physically reasonable glazing that approximates the input entered as well as possible. But the model is not always able to do exactly what is specified when the specifications are not physical.

References¶

Arasteh, D., J.C. Kohler, B. Griffith, Modeling Windows in EnergyPlus with Simple Performance Indices. Lawrence Berkeley National Laboratory. In Draft. Available at

Glazing System Properties¶

The optical properties of a glazing system consisting of N glass layers separated by nonabsorbing gas layers (see Figure 2) are determined by solving the following recursion relations for T\(_{i,j}\), the transmittance through layers i to j; R\(^{f}\)\(_{i,j}\) and R\(^{b}\)\(_{i,j}\), the front and back reflectance, respectively, from layers i to j; and A\(_{j}\), the absorption in layer j. Here layer 1 is the outermost layer and layer N is the innermost layer. These relations account for multiple internal reflections within the glazing system. Each of the variables is a function of wavelength.

In Equation eq:Ajtothefequation, T\(_{i,j}\) = 1 and R\(_{i,j}\) = 0 if i\<0 or j>N.

As an example, for double glazing (N=2), these equations reduce to:

If the above transmittance and reflectance properties are input as a function of wavelength, EnergyPlus calculates "spectral average" values of the above glazing system properties by integrating over wavelength.

The spectral-average solar property is:

The spectral-average visible property is:

where \({E_s}(\lambda )\) is the solar spectral irradiance function and \(V(\lambda )\) is the photopic response function of the eye. The default functions are shown in Table table:solar-spectral-irradiance-function. and Table table:photopic-response-function.. They can be overwritten by user defined solar and/or visible spectrum using the objects Site:SolarAndVisibleSpectrum and Site:SpectrumData. They are expressed as a set of values followed by the corresponding wavelengths for values.

When a choice of Spectral is entered as the optical data type, the correlations to store the glazing system's angular performance are generated based on angular performance at 10 degree increments. When a choice of SpectralAndAngle is entered as the optical data type, the correlations for the glazing system will be generated using 10 degree increments or more if the SpectralAndAngle properties include data for more angles. For each incident angle, the properties of the SpectralAndAngle layer(s) is calculated by linear interpolation, and then the performance of the entire glazing system is calculated for that angle. The glazing system properties at each angle are used to generate polynomial curve fits with 6 coefficients as a function of cosine of incident angle. The polynomial curves are then used in the simulation to calculate optical properties at each timestep.

If a glazing layer has optical properties that are roughly constant with wavelength, the wavelength-dependent values of \(T_{i,i}\), \(R^{f}_{i,i}\) and \(R^{b}_{i,i}\) in Equations eq:Tijequation to eq:Ajtothefequation can be replaced with constant values for that layer.

|| r r r r r r r r r r ||\ \ \ \ , & 9.5, & 42.3, & 107.8, & 181.0, & 246.0, & 395.3, & 390.1, & 435.3, & 438.9,\ 483.7, & 520.3, & 666.2, & 712.5, & 720.7, & 1013.1, & 1158.2, & 1184.0, & 1071.9, & 1302.0,\ 1526.0, & 1599.6, & 1581.0, & 1628.3, & 1539.2, & 1548.7, & 1586.5, & 1484.9, & 1572.4, & 1550.7,\ 1561.5, & 1501.5, & 1395.5, & 1485.3, & 1434.1, & 1419.9, & 1392.3, & 1130.0, & 1316.7, & 1010.3,\ 1043.2, & 1211.2, & 1193.9, & 1175.5, & 643.1, & 1030.7, & 1131.1, & 1081.6, & 849.2, & 785.0,\ 916.4, & 959.9, & 978.9, & 933.2, & 748.5, & 667.5, & 690.3, & 403.6, & 258.3, & 313.6,\ 526.8, & 646.4, & 746.8, & 690.5, & 637.5, & 412.6, & 108.9, & 189.1, & 132.2, & 339.0,\ 460.0, & 423.6, & 480.5, & 413.1, & 250.2, & 32.5, & 1.6, & 55.7, & 105.1, & 105.5,\ 182.1, & 262.2, & 274.2, & 275.0, & 244.6, & 247.4, & 228.7, & 244.5, & 234.8, & 220.5,\ 171.5, & 30.7, & 2.0, & 1.2, & 21.2, & 91.1, & 26.8, & 99.5, & 60.4, & 89.1,\ 82.2, & 71.5, & 70.2, & 62.0, & 21.2, & 18.5, & 3.2  & & &\ \ , & 0.3050, & 0.3100, & 0.3150, & 0.3200, & 0.3250, & 0.3300, & 0.3350, & 0.3400, & 0.3450,\ 0.3500, & 0.3600, & 0.3700, & 0.3800, & 0.3900, & 0.4000, & 0.4100, & 0.4200, & 0.4300, & 0.4400,\ 0.4500, & 0.4600, & 0.4700, & 0.4800, & 0.4900, & 0.5000, & 0.5100, & 0.5200, & 0.5300, & 0.5400,\ 0.5500, & 0.5700, & 0.5900, & 0.6100, & 0.6300, & 0.6500, & 0.6700, & 0.6900, & 0.7100, & 0.7180,\ 0.7244, & 0.7400, & 0.7525, & 0.7575, & 0.7625, & 0.7675, & 0.7800, & 0.8000, & 0.8160, & 0.8237,\ 0.8315, & 0.8400, & 0.8600, & 0.8800, & 0.9050, & 0.9150, & 0.9250, & 0.9300, & 0.9370, & 0.9480,\ 0.9650, & 0.9800, & 0.9935, & 1.0400, & 1.0700, & 1.1000, & 1.1200, & 1.1300, & 1.1370, & 1.1610,\ 1.1800, & 1.2000, & 1.2350, & 1.2900, & 1.3200, & 1.3500, & 1.3950, & 1.4425, & 1.4625, & 1.4770,\ 1.4970, & 1.5200, & 1.5390, & 1.5580, & 1.5780, & 1.5920, & 1.6100, & 1.6300, & 1.6460, & 1.6780,\ 1.7400, & 1.8000, & 1.8600, & 1.9200, & 1.9600, & 1.9850, & 2.0050, & 2.0350, & 2.0650, & 2.1000,\ 2.1480, & 2.1980, & 2.2700, & 2.3600, & 2.4500, & 2.4940, & 2.5370  & & &\

[]

|| r r r r r r r r r r ||\ \ \ \ , & 0.0001, & 0.0001, & 0.0002, & 0.0004, & 0.0006, & 0.0012, & 0.0022, & 0.0040, & 0.0073,\ 0.0116, & 0.0168, & 0.0230, & 0.0298, & 0.0380, & 0.0480, & 0.0600, & 0.0739, & 0.0910, & 0.1126,\ 0.1390, & 0.1693, & 0.2080, & 0.2586, & 0.3230, & 0.4073, & 0.5030, & 0.6082, & 0.7100, & 0.7932,\ 0.8620, & 0.9149, & 0.9540, & 0.9803, & 0.9950, & 1.0000, & 0.9950, & 0.9786, & 0.9520, & 0.9154,\ 0.8700, & 0.8163, & 0.7570, & 0.6949, & 0.6310, & 0.5668, & 0.5030, & 0.4412, & 0.3810, & 0.3210,\ 0.2650, & 0.2170, & 0.1750, & 0.1382, & 0.1070, & 0.0816, & 0.0610, & 0.0446, & 0.0320, & 0.0232,\ 0.0170, & 0.0119, & 0.0082, & 0.0158, & 0.0041, & 0.0029, & 0.0021, & 0.0015, & 0.0010, & 0.0007,\ 0.0005, & 0.0004, & 0.0002, & 0.0002, & 0.0001, & 0.0001, & 0.0001, & 0.0000, & 0.0000, & 0.0000,\ 0.0000  & & & & & & & & &\ \ .380, & .385, & .390, & .395, & .400, & .405, & .410, & .415, & .420, & .425,\ .430, & .435, & .440, & .445, & .450, & .455, & .460, & .465, & .470, & .475,\ .480, & .485, & .490, & .495, & .500, & .505, & .510, & .515, & .520, & .525,\ .530, & .535, & .540, & .545, & .550, & .555, & .560, & .565, & .570, & .575,\ .580, & .585, & .590, & .595, & .600, & .605, & .610, & .615, & .620, & .625,\ .630, & .635, & .640, & .645, & .650, & .655, & .660, & .665, & .670, & .675,\ .680, & .685, & .690, & .695, & .700, & .705, & .710, & .715, & .720, & .725,\ .730, & .735, & .740, & .745, & .750, & .755, & .760, & .765, & .770, & .775,\ .780  & & & & & & & & &\

[]

Calculation of Angular Properties¶

Calculation of optical properties is divided into two categories: uncoated glass and coated glass.

Angular Properties for Uncoated Glass¶

The following discussion assumes that optical quantities such as transmissivity, reflectivity, absorptivity, and index of refraction are a function of wavelength, λ. If there are no spectral data the angular dependence is calculated based on the single values for transmittance and reflectance in the visible and solar range. In the visible range an average wavelength of 0.575 microns is used in the calculations. In the solar range an average wavelength of 0.898 microns is used.

The spectral data include the transmittance, T, and the reflectance, R. For uncoated glass the reflectance is the same for the front and back surfaces. For angle of incidence, \(\phi\) , the transmittance and reflectance are related to the transmissivity, \(\tau\), and reflectivity, \(\rho\), by the following relationships:

The spectral reflectivity is calculated from Fresnel's equation assuming unpolarized incident radiation:

The spectral transmittivity is given by:

The spectral absorption coefficient is defined as:

where \(\kappa\) is the dimensionless spectrally-dependent extinction coefficient and \(\lambda\) is the wavelength expressed in the same units as the sample thickness.

Solving Equation eq:RhoofPhi at normal incidence gives:

Evaluating Equation eq:RofPhiEquation at normal incidence gives the following expression for \(\kappa\):

Eliminating the exponential in Equations eq:TofPhiEquation and eq:RofPhiEquation gives the reflectivity at normal incidence:

where

The value for the reflectivity, \(\rho\)(0), from Equation eq:Rho0Equation is substituted into Equations eq:NAsFunctionOfRhoEquation and eq:KappaAsFunctionOfLambdaRTEquation. The result from Equation eq:KappaAsFunctionOfLambdaRTEquation is used to calculate the absorption coefficient in Equation eq:AlphaAsFunctionOfKappaLambda. The index of refraction is used to calculate the reflectivity in Equation eq:RhoofPhi which is then used to calculate the transmittivity in Equation eq:TauofPhiEquation. The reflectivity, transmissivity and absorption coefficient are then substituted into Equations eq:TofPhiEquation and eq:RofPhiEquation to obtain the angular values of the reflectance and transmittance.

Angular Properties for Coated Glass¶

A regression fit is used to calculate the angular properties of coated glass from properties at normal incidence. If the transmittance of the coated glass is > 0.645, the angular dependence of uncoated clear glass is used. If the transmittance of the coated glass is \(\leq\) 0.645, the angular dependence of uncoated bronze glass is used. The values for the angular functions for the transmittance and reflectance of both clear glass \(({\bar \tau_{clr}},{\bar \rho_{clr}})\)  and bronze glass \(({\bar \tau_{bnz}},{\bar \rho_{bnz}})\) are determined from a fourth-order polynomial regression:

and

The polynomial coefficients are given in Table 4.

Table: Polynomial coefficients used to determine angular properties of coated glass.

These factors are used as follows to calculate the angular transmittance and reflectance:

For T(0) > 0.645:

For T(0) \(\leq\) 0.645:

Calculation of Hemispherical Values¶

The hemispherical value of a property is determined from the following integral:

The integral is evaluated by Simpson's rule for property values at angles of incidence from 0 to 90 degrees in 10-degree increments.

Optical Properties of Window Shading Devices¶

Shading devices affect the system transmittance and glass layer absorptance for short-wave radiation and for long-wave (thermal) radiation. The effect depends on the shade position (interior, exterior or between-glass), its transmittance, and the amount of inter-reflection between the shading device and the glazing. Also of interest is the amount of radiation absorbed by the shading device.

In EnergyPlus, shading devices are divided into four categories, "shades," "blinds," "screens," and "switchable glazing." "Shades" are assumed to be perfect diffusers. This means that direct radiation incident on the shade is reflected and transmitted as hemispherically uniform diffuse radiation: there is no direct component of transmitted radiation. It is also assumed that the transmittance, \(\tau_{sh}\), reflectance, \(\rho_{sh}\), and absorptance, \(\alpha_{sh}\), are the same for the front and back of the shade and are independent of angle of incidence. Many types of drapery and pull-down roller devices are close to being perfect diffusers and can be categorized as "shades."

"Blinds" in EnergyPlus are slat-type devices such as venetian blinds. Unlike shades, the optical properties of blinds are strongly dependent on angle of incidence. Also, depending on slat angle and the profile angle of incident direct radiation, some of the direct radiation may pass between the slats, giving a direct component of transmitted radiation.

"Screens" are debris or insect protection devices made up of metallic or non-metallic materials. Screens may also be used as shading devices for large glazing areas where excessive solar gain is an issue. The EnergyPlus window screen model assumes the screen is composed of intersecting orthogonally-crossed cylinders, with the surface of the cylinders assumed to be diffusely reflecting. Screens may only be used on the exterior surface of a window construction. As with blinds, the optical properties affecting the direct component of transmitted radiation are dependent on the angle of incident direct radiation.

With "Switchable glazing," shading is achieved making the glazing more absorbing or more reflecting, usually by an electrical or chemical mechanism. An example is electrochromic glazing where the application of an electrical voltage or current causes the glazing to switch from light to dark.

Shades and blinds can be either fixed or movable. If movable, they can be deployed according to a schedule or according to a trigger variable, such as solar radiation incident on the window. Screens can be either fixed or movable according to a schedule.

Shades¶

Shade/Glazing System Properties for Short-Wave Radiation¶

Short-wave radiation includes:

• Beam solar radiation from the sun and diffuse solar radiation from the sky and ground incident on the outside of the window,

• Beam and/or diffuse radiation reflected from exterior obstructions or the building itself,

• Solar radiation reflected from the inside zone surfaces and incident as diffuse radiation on the inside of the window,

• Beam solar radiation from one exterior window incident on the inside of another window in the same zone, and

• Short-wave radiation from electric lights incident as diffuse radiation on the inside of the window.

Exterior Shade

For an exterior shade we have the following expressions for the system transmittance, the effective system glass layer absorptance, and the system shade absorptance, taking inter-reflection between shade and glazing into account. Here, "system" refers to the combination of glazing and shade. The system properties are given in terms of the isolated shade properties (i.e., shade properties in the absence of the glazing) and the isolated glazing properties (i.e., glazing properties in the absence of the shade).

Interior Shade

The system properties when an interior shade is in place are the following:

Long-Wave Radiation Properties of Window Shades¶

Long-wave radiation includes:

• Thermal radiation from the sky, ground and exterior obstructions incident on the outside of the window,

• Thermal radiation from other room surfaces incident on the inside of the window, and

• Thermal radiation from internal sources, such as equipment and electric lights, incident on the inside of the window.

The program calculates how much long-wave radiation is absorbed by the shade and by the adjacent glass surface. The system emissivity (thermal absorptance) for an interior or exterior shade, taking into account reflection of long-wave radiation between the glass and shade, is given by:

where \(\rho_{gl}^{lw}\) is the long-wave reflectance of the outermost glass surface for an exterior shade or the innermost glass surface for an interior shade, and it is assumed that the long-wave transmittance of the glass is zero.

The innermost (for interior shade) or outermost (for exterior shade) glass surface emissivity when the shade is present is:

Switchable Glazing¶

For switchable glazing, such as electrochromics, the solar and visible optical properties of the glazing can switch from a light state to a dark state. The switching factor, f\(_{switch}\), determines what state the glazing is in. An optical property, p, such as transmittance or glass layer absorptance, for this state is given by:

where

p\(_{light}\) is the property value for the unswitched, or light state, and p\(_{dark}\) is the property value for the fully switched, or dark state.

The value of the switching factor in a particular time step depends on what type of switching control has been specified: "schedule," "trigger," or "daylighting." If "schedule," f\(_{switch}\) = schedule value, which can be 0 or 1.

Thermochromic Windows¶

Thermochromic (TC) materials have active, reversible optical properties that vary with temperature. Thermochromic windows are adaptive window systems for incorporation into building envelopes. Thermochromic windows respond by absorbing sunlight and turning the sunlight energy into heat. As the thermochromic film warms it changes its light transmission level from less absorbing to more absorbing. The more sunlight it absorbs the lower the light level going through it. Figure 3 shows the variations of window properties with the temperature of the thermochromic glazing layer. By using the suns own energy the window adapts based solely on the directness and amount of sunlight. Thermochromic materials will normally reduce optical transparency by absorption and/or reflection, and are specular (maintaining vision).

On cloudy days the window is at full transmission and letting in diffuse daylighting. On sunny days the window maximizes diffuse daylighting and tints based on the angle of the sun relative to the window. For a south facing window (northern hemisphere) the daylight early and late in the day is maximized and the direct sun at mid day is minimized.

The active thermochromic material can be embodied within a laminate layer or a surface film. The overall optical state of the window at a given time is a function primarily of:

• thermochromic material properties

• solar energy incident on the window

• construction of the window system that incorporates the thermochromic layer

• environmental conditions (interior, exterior, air temperature, wind, etc).

The tinted film, in combination with a heat reflecting, low-e layer allows the window to reject most of the absorbed radiation thus reducing undesirable heat load in a building. In the absence of direct sunlight the window cools and clears and again allows lower intensity diffuse radiation into a building. TC windows can be designed in several ways (Figure 4), with the most common being a triple pane windows with the TC glass layer in the middle a double pane windows with the TC layer on the inner surface of the outer pane or for sloped glazing a double pane with the laminate layer on the inner pane with a low-e layer toward the interior. The TC glass layer has variable optical properties depending on its temperature, with a lower temperature at which the optical change is initiated, and an upper temperature at which a minimum transmittance is reached. TC windows act as passive solar shading devices without the need for sensors, controls and power supplies but their optical performance is dependent on varying solar and other environmental conditions at the location of the window.

EnergyPlus describes a thermochromic window with a Construction object which references a special layer defined with a WindowMaterial:GlazingGroup:Thermochromic object. The WindowMaterial:GlazingGroup:Thermochromic object further references a series of WindowMaterial:Glazing objects corresponding to each specification temperature of the TC layer. During EnergyPlus run time, a series of TC windows corresponding to each specification temperature is created once. At the beginning of a particular time step calculations, the temperature of the TC glass layer from the previous time step is used to look up the most closed specification temperature whose corresponding TC window construction will be used for the current time step calculations. The current time step calculated temperature of the TC glass layer can be different from the previous time step, but no iterations are done in the current time step for the new TC glass layer temperature. This is an approximation that considers the reaction time of the TC glass layer can be close to EnergyPlus simulation time step say 10 to 15 minutes.

Blinds¶

Window blinds in EnergyPlus are defined as a series of equidistant slats that are oriented horizontally or vertically. All of the slats are assumed to have the same optical properties. The overall optical properties of the blind are determined by the slat geometry (width, separation and angle) and the slat optical properties (front-side and back-side transmittance and reflectance). Blind properties for direct radiation are also sensitive to the "profile angle," which is the angle of incidence in a plane that is perpendicular to the window plane and to the direction of the slats. The blind optical model in EnergyPlus is based on Simmler, Fischer and Winkelmann, 1996; however, that document has numerous typographical errors and should be used with caution.

The following assumptions are made in calculating the blind optical properties:

• The slats are flat.

• The spectral dependence of inter-reflections between slats and glazing is ignored; spectral-average slat optical properties are used.

• The slats are perfect diffusers. They have a perfectly matte finish so that reflection from a slat is isotropic (hemispherically uniform) and independent of angle of incidence, i.e., the reflection has no specular component. This also means that absorption by the slats is hemispherically uniform with no incidence angle dependence. If the transmittance of a slat is non-zero, the transmitted radiation is isotropic and the transmittance is independent of angle of incidence.

• Inter-reflection between the blind and wall elements near the periphery of the blind is ignored.

• If the slats have holes through which support strings pass, the holes and strings are ignored. Any other structures that support or move the slats are ignored.

Slat Optical Properties¶

The slat optical properties used by EnergyPlus are shown in the following table.

Table: Slat Optical Properties

It is assumed that there is no direct-to-direct transmission or reflection, so that \({\tau_{dir,dir}} = 0\), \(\rho_{dir,dir}^f = 0\), and \(\rho_{dir,dir}^b = 0\). It is further assumed that the slats are perfect diffusers, so that \({\tau_{dir,dif}}\), \(\rho_{dir,dif}^f\) and \(\rho_{dir,dif}^b\) are independent of angle of incidence. Until the EnergyPlus model is improved to take into account the angle-of-incidence dependence of slat transmission and reflection, it is assumed that \({\tau_{dir,dif}}\) = \({\tau_{dif,dif}}\), \(\rho_{dir,dif}^f\) = \(\rho_{dif,dif}^f\), and \(\rho_{dir,dif}^b\) = \(\rho_{dif,dif}^b\).

Direct Transmittance of Blind¶

The direct-to-direct and direct-to-diffuse transmittance of a blind is calculated using the slat geometry shown in Figure 5(a), which shows the side view of one of the cells of the blind. For the case shown, each slat is divided into two segments, so that the cell is bounded by a total of six segments, denoted by s\(_{1}\) through s\(_{6}\) (note in the following that s\(_{i}\) refers to both segment i and the length of segment i).The lengths of s\(_{1}\) and s\(_{2}\) are equal to the slat separation, h, which is the distance between adjacent slat faces. s\(_{3}\) and s\(_{4}\) are the segments illuminated by direct radiation. In the case shown in Figure 5(a) the cell receives radiation by reflection of the direct radiation incident on s\(_{4}\) and, if the slats have non-zero transmittance, by transmission through s\(_{3}\), which is illuminated from above.

The goal of the blind direct transmission calculation is to determine the direct and diffuse radiation leaving the cell through s\(_{2}\) for unit direct radiation entering the cell through s\(_{1}\).

Direct-to-Direct Blind Transmittance¶

Figure 5(b) shows the case where some of the direct radiation passes through the cell without hitting the slats. From the geometry in this figure we see that

where

Note that we are assuming that the slat thickness is zero. A correction for non-zero slat thickness is described later.

Direct-to-Diffuse Blind Transmittance, Reflectance and Absorptance¶

The direct-to-diffuse and transmittance and reflectance of the blind are calculated using a radiosity method that involves the following three vector quantities:

J\(_{i}\) = the radiosity of segment s\(_{i}\), i.e., the total radiant flux into the cell from s\(_{i}\)

G\(_{i}\) = the irradiance on the cell side of s\(_{i}\)

Q\(_{i}\) = the source flux from the cell side of s\(_{i}\)

Based on these definitions we have the following equations that relate J, G and Q for the different segments:

In addition we have the following equation relating G and J:

where \({F_{ji}}\) is the view factor between \({s_j}\) and \({s_i}\), i.e., \({F_{ji}}\) is the fraction of radiation leaving \({s_j}\) that is intercepted by \({s_i}\).

Using \({J_1} = {Q_1} = 0\) and \({J_2} = {Q_2} = 0\)  and combining the above equations gives the following equation set relating J and Q:

This can be written in the form:

where X is a 4x4 matrix and

We then obtain \(J'\)  from:

The view factors, \({F_{ij}}\), are obtained as follows. The cell we are dealing with is a convex polygon with n sides. In such a polygon the view factors must satisfy the following constraints:

These constraints lead to simple equations for the view factors for n = 3 and 4. For n = 3, we have the following geometry and view factor expression:

For  n = 4  we have:

Applying these to the slat cell shown in Figure 6 we have the following:

The sources for the direct-to-diffuse transmittance calculation are:

For unit incident direct flux, the front direct-to-diffuse transmittance and reflectance of the blind are:

where

and \({J_3}\) to \({J_6}\) are given by Equation eq:QequalsXJprime.

The front direct absorptance of the blind is then:

The direct-to-diffuse calculations are performed separately for solar and visible slat properties to get the corresponding solar and visible blind properties.

Dependence on Profile Angle¶

The direct-to-direct and direct-to-diffuse blind properties are calculated for direct radiation profile angles (see Figure 5) ranging from --90\(^{o}\) to +90\(^{o}\) in 5\(^{o}\) increments. (The "profile angle" is the angle of incidence in a plane that is perpendicular to the window and perpendicular to the slat direction.) In the time step loop the blind properties for a particular profile angle are obtained by interpolation.

Dependence on Slat Angle¶

All blind properties are calculated for slat angles ranging from --90\(^{o}\) to +90\(^{o}\) in 10\(^{o}\) increments. In the time-step loop the slat angle is determined by the slat-angle control mechanism and then the blind properties at that slat angle are determined by interpolation. Three slat-angle controls are available: (1) slat angle is adjusted to just block beam solar incident on the window; (2) slat angle is determined by a schedule; and (3) slat angle is fixed.

Diffuse-to-Diffuse Transmittance and Reflectance of Blind¶

To calculate the diffuse-to-diffuse properties, assuming uniformly distributed incident diffuse radiation, each slat bounding the cell is divided into two segments of equal length (Figure 7), i.e., \({s_3} = {s_4}\) and \({s_5} = {s_6}\).  For front-side properties we have a unit source, \({Q_1} = 1\).  All the other \({Q_i}\) are zero. Using this source value, we apply the methodology described above to obtain G\(_{2}\) and G\(_{1}\). We then have:

The back-side properties are calculated in a similar way by setting Q\(_{2}\) = 1 with the other \({Q_i}\) equal to zero.

The diffuse-to-diffuse calculations are performed separately for solar, visible and IR slat properties to get the corresponding solar, visible and IR blind properties.

Blind properties for sky and ground diffuse radiation¶

For horizontal slats on a vertical window (the most common configuration) the blind diffuse-to-diffuse properties will be sensitive to whether the radiation is incident upward from the ground or downward from the sky (Figure 8). For this reason we also calculate the following solar properties for a blind consisting of horizontal slats in a vertical plane:

\(\tau_{bl,f}^{gnd - dif,dif} = {\rm{ }}\) front transmittance for ground diffuse solar

\(\tau_{bl,f}^{sky - dif,dif} = {\rm{ }}\) front transmittance for sky diffuse solar

\(\rho_{bl,f}^{gnd - dif,dif} =\) front reflectance for ground diffuse solar

\(\rho_{bl,f}^{sky - dif,dif} = {\rm{ }}\) front reflectance for sky diffuse solar

\(\alpha_{bl,f}^{gnd - dif,dif} = {\rm{ }}\) front absorptance for ground diffuse solar

\(\alpha_{bl,f}^{sky - dif,dif} = {\rm{ }}\) front absorptance for sky diffuse solar

These are obtained by integrating over sky and ground elements, as shown in Figure 8, treating each element as a source of direct radiation of irradiance \(I({\phi_s})\) incident on the blind at profile angle \({\phi_s}\). This gives:

We assume that the sky radiance is uniform. This means that \({I_{sky}}\) is independent of \({\phi_s}\), giving:

The corresponding ground diffuse quantities are obtained by integrating \({\phi_s}\) from \(- \pi /2\) to 0.

An improvement to this calculation would be to allow the sky radiance distribution to be non-uniform, i.e., to depend on sun position and sky conditions, as is done in the detailed daylighting calculation (see "Sky Luminance Distributions" under "Daylight Factor Calculation").

Correction Factor for Slat Thickness¶

A correction has to be made to the blind transmittance, reflectance and absorptance properties to account for the amount of radiation incident on a blind that is reflected and absorbed by the slat edges (the slats are assumed to be opaque to radiation striking the slat edges). This is illustrated in Figure 9 for the case of direct radiation incident on the blind. The slat cross-section is assumed to be rectangular. The quantity of interest is the fraction, f\(_{edge}\), of direct radiation incident on the blind that strikes the slat edges. Based on the geometry shown in Figure 9 we see that

The edge correction factor for diffuse incident radiation is calculated by averaging this value of f\(_{edge}\) over profile angles, \(\varphi_s\), from -90\(^{o}\) to +90\(^{o}\).

As an example of how the edge correction factor is applied, the following two equations show how blind front diffuse transmittance and reflectance calculated assuming zero slat thickness are modified by the edge correction factor. It is assumed that the edge transmittance is zero and that the edge reflectance is the same as the slat front reflectance, \(\rho_f\).

Comparison with ISO 15099 Calculation of Blind Optical Properties¶

Table 6 compares EnergyPlus and ISO 15099 [2001] calculations of blind optical properties for a variety of profile angles, slat angles and slat optical properties. The ISO 15099 calculation method is similar to that used in EnergyPlus, except that the slats are divided into five equal segments. The ISO 15099 and EnergyPlus results agree to within 12%, except for the solar transmittances for the 10-degree slat angle case. Here the transmittances are small (from 1% to about 5%) but differ by about a factor of up to two between ISO 15099 and EnergyPlus. This indicates that the slats should be divided into more than two segments at small slat angles.

| Slat Properties | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | N\(_{Basis}\) | Number of elements in the (incoming or outgoing) basis | | | | | | | | | | | N\(_{Sun}\) | Number of basis directions that may be sun directions (depends on fenestration orientation | | | | | | | | | | | N\(^{(Gnd)}_{Sun}\) | Number of sun directions that give significantly different ground irradiation conditions, as seen by fenestration | | | | | | | | | | | N\(_{Sf}\) | Number of reflecting surfaces viewable by fenestration (depends on fenestration orientation) | | | | | | | | | | | N\(^{(Sf,n)}_{Sun}\) | Number of time steps for which surface n is sunlit (depends on orientation of surface n; determined during shading calculation) | | | | | | | | | | | N\(^{(Sf,n)}_{Refl}\) | Number of basis directions that may be reflected sun directions from surface n (depends on orientation of fenestration and surface n). | | | | | | | | | | | N\(_{IntSurf}\) | Number of interior surfaces in the zone containing the fenestration | | | | | | | | | | | N\(_{Layers}\) | Number of thermal layers in the fenestration system | | | | | | | | | | | Arrays | | | | | | | | | | | | \(\mathcal{A}\)\(^{F,l}_{i}\), \(\mathcal{A}\)\(^{B,l}_{i}\) | Absorptance vector element; N\(_{Basis}\) | | | | | | | | | | | T\(_{ij}\) | Transmittance matrix element; N\(_{Basis}\) X N\(_{Basis}\) | | | | | | | | | | | V\(^{(Sky)}_{i}\) | Sky viewed fraction; one-dimensional, N\(_{Basis}\) | | | | | | | | | | | V\(^{(Sf,n)}_{i}\) | Fraction of surface n viewed; N\(_{Basis}\) X N\(_{Sf}\) | | | | | | | | | | | \(\rho\)\(^{(n)}\), \(\rho\)\(^{(sp,n)}\) | Surface n diffuse, specular reflectance; N\(_{Sf}\) (already stored by E+) | | | | | | | | | | | U\(^{(D,n)}_{i\,Sun(tsh)}\) | Fraction of the image of \(\Delta\Omega\)\(_{i}\) on surface n that views the sun when it is in direction Sun(tsh); N\(_{Basis}\) X N\(_{Sf}\) X N\(^{(Sf,n)}_{Sun}\) | | | | | | | | | | | V\(^{(Gnd)}_{i}\) | Fraction of \(\Delta\Omega\)\(_{i}\) that views ground; N\(_{Basis}\) | | | | | | | | | | | U\(^{(Sky, Gnd)}_{i}\) | Fraction of sky radiation received by the image of \(\Delta\Omega\)\(_{i}\) on the ground; N\(_{Basis}\) | | | | | | | | | | | U\(^{(D,Gnd)}_{i\,Sun(tsh)}\) | Fraction of direct solar radiation for sun direction Sun(tsh) received by image of \(\Delta\Omega\)\(_{i}\) on ground; N\(^{Gnd}_{Sim}\) X N\(_{Basis}\) | | | | | | | | | | | V\(^{(D)}_{i\,s(t)}\) | Fraction of fenestration area irradiated by direct solar radiation for direction i, given that sun angle is s(t); N\(_{Sun}\) X N\(_{Basis}\) | | | | | | | | | | | F\(^{(k)}_{j}\) | Fraction of radiation in direction j leaving fenestration interior that arrives at surface k; N\(_{Basis}\) X N\(_{IntSurf}\) | | | | | | | | | | | Z\(^{(Sky), k}\) | Sky irradiation factor; N\(_{IntSurf}\) | | | | | | | | | | | Z\(^{(sp,Sf,n),k}_{r(t)}\) | Exterior surface specular irradiation factor; N\(^{(Sf,n)}_{Sun}\) X N\(_{Sf}\) X N\(_{IntSurf}\) | | | | | | | | | | | Z\(^{(Sun,Sf,n),k}_{s(t)}\) | Exterior surface direct-diffuse irradiation factor; N\(^{(Sf,n)}_{Sun}\) X N\(_{Sf}\) X N\(_{IntSurf}\) | | | | | | | | | | | Z\(^{(Sky,Sf,n),k}\) | Exterior surface sky irradiation factor; N\(_{Sf}\) X N\(_{IntSurf}\) | | | | | | | | | | | Z\(^{(D,Gnd),k}_{s(t)}\) | Ground-reflected direct solar irradiation factor (given sun direction s(t)); N\(^{(Gnd)}_{Sun}\) X N\(_{IntSurf}\) | | | | | | | | | | | Z\(^{(Sky,Gnd),k}\) | Ground-reflected diffuse solar irradiation factor; N\(_{IntSurf}\) | | | | | | | | | | | Z\(^{(Sun),k}_{s(t)}\) | Direct solar irradiation factor; N\(_{Sun}\) X N\(_{IntSurf}\) | | | | | | | | | | | K\(^{(Sky),l}\) | Sky absorption factor; N\(_{Layers}\) | | | | | | | | | | | K\(^{(sp,Sf,n),l}_{r(t)}\) | Exterior surface specular absorption factor; N\(_{Sf}\) X N\(^{(Sf,n)}_{Refl}\) X N\(_{Layers}\) | | | | | | | | | | | K\(^{(Sun,Sf,n),l}_{s(t)}\) | Exterior surface diffusely reflected direct sun absorption factor; N\(_{Sf}\) X N\(^{(Sf,n)}_{Sun}\) X N\(_{Layers}\) | | | | | | | | | | | K\(^{(Sky,Sf,n),l}\) | Exterior surface reflected sky radiation absorption factor; N\(_{Sf}\) X N\(_{Layers}\) | | | | | | | | | | | K\(^{(D,Gnd),l}_{s(t)}\) | Ground-reflected direct solar absorption factor; N\(^{(Gnd)}_{Sun}\) X N\(_{Layers}\) | | | | | | | | | | | K\(^{(Sky,Gnd),l}\) | Ground-reflected sky radiation absorption factor; N\(_{Layers}\) | | | | | | | | | | | K\(^{(Sun),l}_{s(t)}\) | Direct sunlight absorption factor; N\(_{Sun}\) X N\(_{Layers}\) | | | | | | | | | |

Table: Comparison of blind optical properties calculated with the EnergyPlus and ISO 15099 methods. EnergyPlus values that differ by more than 12% from ISO 15099 values are shown in bold italics.

Absorption¶

For thermal calculations, it is necessary to know the energy absorbed in each layer of the fenestration.  This depends only on the incident geometry, but otherwise is calculated in the same manner as the solar flux incident on interior surfaces.  For a given layer l of a fenestration f, we define a source-referenced absorption factor, K(source),l.  This is the amount of energy absorbed in layer l divided by the relevant solar intensity (which might be beam, diffuse, or reflected beam or diffuse, depending on the source of the radiation).  These absorption factors and the resultant source-specific absorbed solar powers are calculated by the analogs [see Equation eq:QnVectorMatrixEquation or Equations eq:ZSkykEquation through eq:ZstSunkEquation]:

Comment on Bases¶

Use of the basis in the above discussion has been mostly implicit, but it should nevertheless be clear that the essential feature of the basis is that it is a two-element list (i.e., a 2 X N array): it associates with an incident (i) or outgoing (j) direction index a vector pi (or pj ) that is a unit vector giving the direction of the radiation, the specification of which is two angles in some coordinate system.  The incident and outgoing bases of course must match the matrix elements of the fenestration properties.  These bases will (certainly in the case of WINDOW program; probably in the case of other input sources) have a structure: ordering of the elements, etc.  However, after the initialization of the hourly loop calculation, this structure will be irrelevant:  EnergyPlus will retain only those incoming and outgoing directions that are essential to the calculation with (one would hope, most of) the others combined into irradiation factors.  At this point, the basis will truly be an arbitrary list.  It follows that the specification of the basis in the EnergyPlus input should be determined by (1) the source of fenestration property data, and (2) user convenience.

A related point concerns the specification of a basis for specular glazings, i.e., multiple layers of glass.  These fenestrations are both specular (input direction = output direction) and axially symmetric.  These properties have different effects on the calculation.

The specular property means that one should not be using Equation eq:SofptotheTDiffEq at all to describe the transmittance.  Instead, one should use the equation:

This equation is shoehorned into the integral calculation of equation through the use of a delta function in the incident direction vector, resulting (after the discretization) in a diagonal matrix for the transmittance (or reflectance).  The outgoing radiance element on the diagonal would be calculated as T\(_{ii}\)\(\Lambda\)\(_{ii}\), where multiplication by \(\Lambda\)\(_{ii}\) substitutes for integration over the basis solid angle element. For a specular glazing, \(T_{ii} = \tau(\bf{p}_i^{(T)})/\Lambda_{ii}\), so one recovers the correct transmittance when one does the multiplication. However, there is still a problem in principle: For a specular fenestration, the angular spread of the outgoing radiation will be that of the source, which for direct sunlight is very small; the calculation, however, assumes the angular spread of the basis element. This problem disappears in the geometric approximation to be used in EnergyPlus: by considering only the central direction of each basis element, the outgoing radiation in that direction is essentially assumed to be specular, so the blurring in the discretization is undone.

The axial symmetry of conventional glazings means that the transmittance (or reflectance) depends on only the incident angle, not the azimuthal angle about the normal to the fenestration plane.  So if one specifies the diagonal elements of the matrix, all of the terms with the same incident angles but different azimuthal angles will be the same.  One could alternatively specify only the specular transmittance at each of the incident angle values, provided one also indicated that it was for an axially symmetric fenestration.  Since expanding this set of values to the equivalent diagonal elements is a trivial calculation, how one specifies a specular glazing is completely a question of user convenience.  For example, if one were dealing with the WINDOW full basis, would it be more user-friendly to specify

1. \(T_{ii} = \tau(\bf{p}_i^{(T)})/\Lambda_{ii}\) , for 145 values, 135 of which are repeats of the previous value

2. \(\tau ({\theta_i})\)  for 9 values of incident angle, \(\theta\)\(_{i}\) ?

Interior Solar Radiation Transmitted by Complex Fenestration¶

Diffuse Solar Radiation Transmitted by Complex Fenestration

Distribution of solar radiation transmitted through exterior window is divided on diffuse and direct part.

Diffuse solar transmitted through exterior complex fenestration and absorbed in interior walls is calculated and treated in same way as described in the section on Initial Distribution of Diffuse Solar Transmitted through Exterior and Interior Windows. Even though that BSDF is given for various directions, for purpose of diffuse solar radiation, transmittance and reflectances of fenestration system is integrated over incoming and outgoing hemisphere.  Because incoming diffuse solar radiation is divided on ground and sky parts, integration of incoming hemisphere is also performed over ground and sky part (see Equation eq:SSfnpTEquation).

Direct Solar Radiation Transmitted by Complex Fenestration

Direct solar (beam) transmitted through exterior window is using same overlap calculations (see Figure fig:vertical-section-through-a-two-zone-building) for each outgoing basis direction.  For certain sun position, algorithm calculates equivalent incoming beam number.  The inside beam solar irradiance is calculated in similar manner as described in the section titled Interior Beam Radiation.

i = exterior window number

N\(_{extwin}\) = number of exterior windows in zone

N\(_{out}\) = Beam number of exterior windows in zone

CosInc\(_{i}\) = cosine of angle of incidence of beam on exterior window i

TBm\(_{k,j}\) = beam-to-beam transmittance of exterior window i at incidence direction k outgoing direction j

Λ\(_{k,j}\) = lambda value of exterior window i at incidence direction k for outgoing direction j

Aoverlap\(_{k,j}\)(SurfNum) = beam solar irradiated area of surface SurfNum projected back onto the plane of exterior window i for incoming direction k and outgoing direction j (the Aoverlap's for an exterior window sum up to the glazed area of the window)

AbsIntSurf(SurfNum) = inside face solar absorptance of surface SurfNum

A(SurfNum) = area of surface SurfNum [m\(^{2}\)]

Equation eq:AISurfEquation is valid as long as surface which is hit by transmitted solar radiation is not another complex fenestration.  In that case, for beam which is transmitted from other exterior window and reaches back surface of this window, angle of incidence needs to be taken into account.

Interior Solar Absorbed by Complex Fenestration¶

Solar radiation absorbed in window layers is coming from three sources: Diffuse radiation from sky and ground, direct radiation from the sun and beam radiation coming from the sun and it is transmitted through other exterior windows.

Diffuse Radiation from Sky and Ground

Energy absorbed in the layers and which originates from diffuse radiation from sky and ground is represented by following equation:

where

WinSkyFtAbs(Lay) = front absorptance averaged over sky for layer (Lay) and window belonging to Surf

WinSkyGndAbs(Lay) = front absorptance averaged over ground for layer (Lay) and window belonging to Surf

SkySolarInc = incident diffuse solar from the sky

GndSolarInc = incident diffuse solar from the ground

Direct Radiation from the Sun

Energy absorbed in the layers and which originates from direct solar radiation is given by following equation:

where

AWinSurf(SurfNum,Lay) -- is time step value of factor for beam absorbed in fenestration glass layers

BeamSolar -- Current beam normal solar irradiance

Factor for time step value is given by equation:

where

WinBmFtAbs(Lay,HourOfDay,TimeStep) -- is front directional absorptance for given layer and time

CosInc -- cosine of beam solar incident angle

SunLitFract -- sunlit fraction without shadowing effects of frame and divider

OutProjSLFracMult(HourOfDay) - Multiplier on sunlit fraction due to shadowing of glass by frame and divider outside projections.

Direct Solar Radiation Coming from Sun and it is Transmitted Through Other Windows

Direct solar radiation transmitted through other windows is using solar overlap calculations described in the section on Overlapping Shadows. Overlapping is used to determine amount of energy transferred through the window is hitting certain surface.  That is used to calculate energy absorbed in walls and same approach will be used to calculate energy absorbed in window layers (Equation eq:AISurfEquation).  In case when receiving surface is complex fenestration, it is not enough just to apply Equation eq:AISurfEquation because factor AbsIntSurf is now depending of incoming angle which is defined through front and back directional absorptance matrices.  It would mean that for each outgoing directions of transmitting complex fenestration, algorithm would need to determine what is best matching basis direction of receiving surface.  Best receiving direction is used to determine absorptance factors which will be used in Equation eq:AISurfEquation.  It is important to understand that for basis definition, each unit vector defining one beam is going towards surface, which would mean that best matching directions from surface to surface will actually have minimal dot product.

where

Best\(_{in}\) -- is best matching receiving direction basis dot product (in\(_{k}\))

out\(_{p}\) -- current transmitting complex fenestration direction

in\(_{1}\), ..., in\(_{N}\) -- set of receiving complex fenestration basis directions

Result of Equation eq:BestinEquation is minimal dot product, which corresponds to best matching direction of receiving surface.  If we mark that direction with index k, then Equation eq:AISurfEquation becomes:

where

AbsIntSurf\(_{k}\)(SurfNum) -- directional absorptance for the receiving surface for the best matching direction.

Everything else is same as described in Equation eq:AISurfEquation.

References¶

Klems, J. H. 1994A. "A New Method for Predicting the Solar Heat Gain of Complex Fenestration Systems: I. Overview and Derivation of the Matrix Layer Calculation.". ASHRAE Transactions. 100(pt.1): 1073-1086.

Klems, J. H. 1994B. "A New Method for Predicting the Solar Heat Gain of Complex Fenestration Systems: II. Detailed Description of the Matrix Layer Calculation.". ASHRAE Transactions. 100(pt.1): 1073-1086.

Klems, J. H. 1995. "Measurements of Bidirectional Optical Properties of Complex Shading Devices.". ASHRAE Transactions. 101(pt 1; Symposium Paper CH-95-8-1(RP-548)): 791-801.

Klems, J. H. 1996. "A Comparison between Calculated and Measured SHGC for Complex Glazing Systems.". ASHRAE Transactions. 102(Pt. 1; Symposium Paper AT-96-16-1): 931-939.

Klems, J. H. 1996. "Calorimetric Measurements of Inward-Flowing Fraction for Complex Glazing and Shading Systems.". ASHRAE Transactions. 102(Pt. 1; Symposium Paper AT-96-16-3): 947-954.

Papamichael, K. J. 1998. "Determination and Application of Bidirectional Solar-Optical Properties of Fenestration Systems.". Cambridge, MA: 13^th National Passive Solar Conference.